Brandão and Svore [BS16] very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m ≈ n, which is the same as classical.
Additional Metadata
Keywords Linear programs, Lower bounds, Quantum algorithms, Semidefinite programs
Stakeholder QuSoft
Persistent URL dx.doi.org/10.1109/FOCS.2017.44
Conference Annual IEEE Symposium on Foundations of Computer Science
Grant This work was funded by the European Commission 7th Framework Programme; grant id erc/615307 - Progress in quantum computing: Algorithms, communication, and applications (QPROGRESS), This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/617.001.351 - Approximation Algorithms, Quantum Information and Semidefinite Optimization
Citation
van Apeldoorn, J, Gilyén, A.P, Gribling, S.J, & de Wolf, R. M. (2017). Quantum SDP-solvers: Better upper and lower bounds. In FOCS (pp. 403–414). doi:10.1109/FOCS.2017.44